This proof is an expanded version of the argument outlined in [1]}.
By Lemma REF  below and the fact that \(\widetilde{Y}_{(1)} \ncong Y\)  (see Remark REF ), it suffices to show that if \(\tilde{z}\in \widetilde{Z}_{(1),{\min }}^{ss}\)  lies over \(z \in C(Z^{ss}_{\min },U)\) , then \(\tilde{z}\)  is not fixed by \(U^{\prime }:= \operatorname{Stab}_{U}(z)\) . We have
\( Z_{\min }^{U^{\prime }} \subset C(Z_{\min },U,d_{\max }) \subset Z_{\min }. \) 
By embedding \(Y\)  in a projective space, we can assume without loss of generality that \(Y = \mathbb {P}^m\)  and as \(Z_{\min }^{U^{\prime }}\)  and \(Z_{\min }\)  are linear subspace we can assume we have taken coordinates so there are \(0 \le n \le m^{\prime } \le m\)  such that \(Z_{\min }^{U^{\prime }}\)  (resp. \(Z_{\min }\) ) is the vanishing locus of the first \(m^{\prime }\)  (resp. n) coordinates and \(p([x_0: \cdots : x_m]) = [0 : \cdots :0 : x_{n} : \cdots : x_m]\) . We claim the \(U^{\prime }\) -action on \(\mathbb {P}^m\)  with respect to the blocks \(n \le m^{\prime } \le m\)  has the form
\( u^{\prime } \mapsto \left( \begin{array}{ccc} A(u^{\prime }) & B(u^{\prime }) & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{array} \right). \) 
Indeed, the final column has this form, as \(U^{\prime }\)  acts trivially on \(Z_{\min }^{U^{\prime }}\) . With respect to the two blocks given by \(n \le m\)  (given by the minimal \(\lambda \) -weight), this matrix is block upper triangular, as \(\lambda \)  grades \(U\) . Finally, the middle column has this form, as \(p(u \cdot x) = x\)  for \(x \in Z_{\min }\) .
Since the exceptional divisor is given by the projectivised normal bundle, \(\tilde{z}\)  is represented by some \([\xi ] \in \mathbb {P}(T_zZ_{\min } / T_z(Z_{\min }^{U^{\prime }}))=\pi _{(1)}^{-1}(z)\) . We can take coordinates so \(z = [0: \cdots :0 :1]\)  and then take local coordinate \(x_j/x_m\)  around \(z\) . From the above block form of the action, one sees for any \(\xi \in T_zZ_{\min } \setminus T_z(Z_{\min }^{U^{\prime }})\) ,
there is a \(u^{\prime } \in U^{\prime }\)  such that \(u^{\prime } \xi - \xi \notin T_zZ_{\min }\) . Indeed, in the above representation of the \(U^{\prime }\) -action, we have some \(u^{\prime }\)  where \(B(u^{\prime }) \ne 0\) , as \(Z_{\min }^{U^{\prime }} \ne Z_{\min }\) . Therefore \(\tilde{z}\) , which is represented by some \([\xi ]\) , is not fixed by \(U^{\prime }\) .
